Greedy-based approaches for solving nonoverlapping, unordered, type-2 square jigsaw puzzles in the literature, primarily consider single-side compatibility, which increases the number of false-positive candidate neighbors. In the assembly phase, these solvers generally resolve the issue by considering all possible combinatorial combinations of the candidate neighbors to determine an optimal match, increasing the solver's computational complexity. To address the same, this paper proposes an efficient greedy asymmetric assembly strategy with two-side verification to solve square jigsaw puzzles. More precisely, the proposed greedy strategy constructs nonoverlapping blocks via unambiguous jigsaw neighbors, such that at least two sides are always connected to the same block. The idea is to reconstruct the puzzle by optimally growing the subset of the local optimal solutions. The implementation leverages a disjoint set data structure to grow and merge blocks efficiently, achieving a space complexity of O (N) and an amortized time complexity of O (N log₂ N). Extensive experimental evaluations validate the solver's effectiveness across standard datasets via direct, neighbor, and perfect compatibility matrices. Furthermore, the algorithm demonstrates its versatility in handling diverse type-2 scenarios, including small puzzle sizes, missing pieces, and mixed puzzles. Our method achieves near-perfect precision in the construction of initial jigsaw blocks, with more than 74% sides of the total jigsaw pieces, as demonstrated by experimental analysis.